Properties

Label 2-979-979.10-c0-0-0
Degree $2$
Conductor $979$
Sign $-0.915 + 0.402i$
Analytic cond. $0.488584$
Root an. cond. $0.698988$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.142 − 0.0101i)3-s + (−0.654 − 0.755i)4-s + (−0.368 − 1.25i)5-s + (−0.969 + 0.139i)9-s + (−0.959 − 0.281i)11-s + (−0.100 − 0.100i)12-s + (−0.0653 − 0.175i)15-s + (−0.142 + 0.989i)16-s + (−0.708 + 1.10i)20-s + (−0.574 + 0.767i)23-s + (−0.601 + 0.386i)25-s + (−0.275 + 0.0600i)27-s + (−1.12 − 1.50i)31-s + (−0.139 − 0.0303i)33-s + (0.740 + 0.641i)36-s + (1.41 − 1.41i)37-s + ⋯
L(s)  = 1  + (0.142 − 0.0101i)3-s + (−0.654 − 0.755i)4-s + (−0.368 − 1.25i)5-s + (−0.969 + 0.139i)9-s + (−0.959 − 0.281i)11-s + (−0.100 − 0.100i)12-s + (−0.0653 − 0.175i)15-s + (−0.142 + 0.989i)16-s + (−0.708 + 1.10i)20-s + (−0.574 + 0.767i)23-s + (−0.601 + 0.386i)25-s + (−0.275 + 0.0600i)27-s + (−1.12 − 1.50i)31-s + (−0.139 − 0.0303i)33-s + (0.740 + 0.641i)36-s + (1.41 − 1.41i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(979\)    =    \(11 \cdot 89\)
Sign: $-0.915 + 0.402i$
Analytic conductor: \(0.488584\)
Root analytic conductor: \(0.698988\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{979} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 979,\ (\ :0),\ -0.915 + 0.402i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4965431162\)
\(L(\frac12)\) \(\approx\) \(0.4965431162\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.959 + 0.281i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
good2 \( 1 + (0.654 + 0.755i)T^{2} \)
3 \( 1 + (-0.142 + 0.0101i)T + (0.989 - 0.142i)T^{2} \)
5 \( 1 + (0.368 + 1.25i)T + (-0.841 + 0.540i)T^{2} \)
7 \( 1 + (-0.540 - 0.841i)T^{2} \)
13 \( 1 + (0.989 - 0.142i)T^{2} \)
17 \( 1 + (-0.654 + 0.755i)T^{2} \)
19 \( 1 + (-0.281 - 0.959i)T^{2} \)
23 \( 1 + (0.574 - 0.767i)T + (-0.281 - 0.959i)T^{2} \)
29 \( 1 + (0.540 + 0.841i)T^{2} \)
31 \( 1 + (1.12 + 1.50i)T + (-0.281 + 0.959i)T^{2} \)
37 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
41 \( 1 + (0.989 + 0.142i)T^{2} \)
43 \( 1 + (0.540 - 0.841i)T^{2} \)
47 \( 1 + (-0.817 + 0.708i)T + (0.142 - 0.989i)T^{2} \)
53 \( 1 + (1.45 + 1.25i)T + (0.142 + 0.989i)T^{2} \)
59 \( 1 + (1.19 + 0.0855i)T + (0.989 + 0.142i)T^{2} \)
61 \( 1 + (0.909 - 0.415i)T^{2} \)
67 \( 1 + (-1.19 + 1.37i)T + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (-0.234 + 0.797i)T + (-0.841 - 0.540i)T^{2} \)
73 \( 1 + (0.959 + 0.281i)T^{2} \)
79 \( 1 + (-0.959 - 0.281i)T^{2} \)
83 \( 1 + (0.755 + 0.654i)T^{2} \)
97 \( 1 + (0.540 - 0.158i)T + (0.841 - 0.540i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.476774113083334819629797635521, −9.188764137594333910421004731785, −8.170157381277718640638822820839, −7.75681189109484420347087280810, −5.99344885597816822152327846161, −5.48575171383093749403486680704, −4.69104334451561357836108718663, −3.69394101440623208029756822572, −2.11391224914818275159478963833, −0.45068107901594621129687256700, 2.65060952036104922893054858440, 3.14418331887579546734207469561, 4.23688357522998897329022960200, 5.32691809646948707714805912604, 6.44556580984062698492870388461, 7.38548613333589462323419311552, 8.030864127688147162903681312545, 8.755346459028197276288320273431, 9.737517322146844467390200562704, 10.64043260631488140979156318913

Graph of the $Z$-function along the critical line